Answer:Increasing in x∈(0,π/4)∪(5π/4,2π) decreasing in(π/4,5π/4)
Step-by-step explanation:
given f(x) = sin(x) + cos(x)
f(x) can be rewritten as ![\sqrt{2} [\frac{sin(x)}{\sqrt{2} }+\frac{cos(x)}{\sqrt{2} } ]..................(a)\\\\\ \frac{1}{\sqrt{2} } = cos(45) = sin(45)\\\\](https://tex.z-dn.net/?f=%5Csqrt%7B2%7D%20%5B%5Cfrac%7Bsin%28x%29%7D%7B%5Csqrt%7B2%7D%20%7D%2B%5Cfrac%7Bcos%28x%29%7D%7B%5Csqrt%7B2%7D%20%7D%20%20%5D..................%28a%29%5C%5C%5C%5C%5C%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%20%7D%20%3D%20cos%2845%29%20%3D%20sin%2845%29%5C%5C%5C%5C)
Using these result in equation a we get
f(x) = ![\sqrt{2} [ cos(45)sin(x)+sin(45)cos(x)]\\\\= \sqrt{2} [sin(45+x)]..........(b)](https://tex.z-dn.net/?f=%5Csqrt%7B2%7D%20%5B%20cos%2845%29sin%28x%29%2Bsin%2845%29cos%28x%29%5D%5C%5C%5C%5C%3D%20%5Csqrt%7B2%7D%20%5Bsin%2845%2Bx%29%5D..........%28b%29)
Now we know that for derivative with respect to dependent variable is positive for an increasing function
Differentiating b on both sides with respect to x we get
f '(x) = 
where x∈(0,2π)
we know that cox(x) > 0 for x∈[0,π/2]∪[3π/2,2π]
Thus for cos(π/4+x)>0 we should have
1) π/4 + x < π/2 => x<π/4 => x∈[0,π/4]
2) π/4 + x > 3π/2 => x > 5π/4 => x∈[5π/4,2π]
from conditions 1 and 2 we have x∈(0,π/4)∪(5π/4,2π)
Thus the function is decreasing in x∈(π/4,5π/4)
4.83
answer above answer above answer above
Answer:
(a)z=-1
(b)48.5 pounds
(c)61.5 pounds
Step-by-step explanation:
Given:
Mean, μ = 55 pounds
Standard deviation,σ = 6 pounds.
(a)For a dog that weighs 49 pounds.
x=49 pounds
The z-score
(b)When a dog has a z-score of -1.09
The weight of a dog with a z-score of -1.09 is 48.5 pounds.
(c)When a dog has a z-score of 1.09
The weight of a dog with a z-score of 1.09 is 61.5 pounds.
Answer:
d
Step-by-step explanation:
Answer:
All real numbers; the simplification is described below.
Step-by-step explanation:
5(- 3x - 2) - (x - 3) = -4(4x + 5) + 13
Let's keep in mind PEMDAS (the order of operations) to simplify and solve this. First, we'll distribute inside each set of parenthesis!
5(-3x) + 5(-2) -1(x) -1(-3) = -4(4x) -4(5) + 13
See how we've taken the outer term on the outside of the parenthesis and multiplied it to each term inside the parenthesis. Simplify further.
-15x - 10 - x + 3 = -16x - 20 + 13
Now we can combine like terms! I will bold the terms I am combining.
-15x - 10 - x + 3 = -16x - 20 + 13 (combining the x terms on the left)
-16x - 10 + 3 = -16x - 20 + 13 (combining the constants on both sides)
-16x - 7 = -16x - 7
Aha! It looks like we have an equation that's literally equal to itself. If we were to simplify this, or put in any number, we'd get that number equals itself. For example:
-16(1) - 7 = -16(1) - 7
-16 - 7 = -16 - 7
-21 = -21
Any number you put in here will make the equation true, so these lines must be the same line. That means the solution is all real numbers.
